Project Involving Newton's Cradle

We are analyzing a simple Newton's Cradle in class to supplement our learning of impulse and momentum. Our project requires us to determine the amount of time it takes for the apparatus to come to rest. In our analysis we are to only use two masses, not the typical 3+. My group and I are trying to use the harmonic motion of a simple pendulum with each mass independent of the other. Will this simplify our task or only make it harder? As of now, there are NO knowns. I have attached our excel file. Our next step should be to analyze the collision, but we are not sure how.

It is not clear whether you are including the effects of damping, but if you aren't then you should, because if you don't your model will never reach a state of rest (i.e. the system will keep oscillating forever).

Next step is to identify the sources of damping. Two that spring to mind are;
- Air resistance.
- Inelastic collisions.

Air resistance can probably be neglected in this case.

On the issue of treating the masses independantly - You are trying to determine how long it will be before the system comes to rest. Essentially you are measuring the rate at which the system dissipates energy. Treating the masses seperately, or indeed trying to use forces at all would be taking the difficult path. My suggestion would be to relate energy loss per second to the number of collisions per second, which can in turn be related to the natural frequency of the pendulum.

Essentially you can treat both masses a a single pendulum that periodically loses energy.

As you mentioned, there are no knowns, so there won't be a single definative answer. You should obtain an answer that depends on the initial height of the pendulum the energy lost in each collision (which could be further related to the elasticity of the balls themselves etc depending on how in depth you wished to go), and the natural frequency of the pendulum (which depends on string length).

Thank you for your responses. Mr. James, we do realize that we need to determine energy loss, but there are no tools for measurement in this project. We did consinder energy relationships, but they seemed to only further complicate our problem since we cannot measure the height and angle of the second mass after the collision. Mr. Claude, we have not yet included damping, but that is the are we are trying to solve. Thank you for clarifying my question. We are not treating the masses seperately but rather assuming the two masses to be a single pendulum with the collision being the damping effect. At the moment we are looking at using momentum-impulse relations to analyze the actual collision. We have a relationship between the two velocities and the coefficient of restitution, but we are not sure how to get that back to the harmonic motion in the excel file.

Okay, I took a look at the excel file and I will make some further comments.

Do you think it is fully necessary to model the position, velocity and acceleration of the pendulum? All you are trying to figure out is how long it will take to lose all its energy.

Also, I think there is some confusion between the angular velocity of the pendulum and the angular velocity of its oscillation. The angular velocity will vary, reaching a maximum at the lowest point and zero at the endpoints. The angular velocity of the oscillation will be constant because the period (as I noticed you have calculated) is constant, since g and L are constant. The angular velocity of the oscillation is defined as;

[tex] \omega = \frac{2\pi}{T} [/tex]

It is this quantity that is constant. To avoid confusion, I suggest using v rather than omega to denote the velocity, since by the small angle approximation these two quanities are nearly equal anyway.

Finally, as I mentioned in my previous post, the energy lost per collision does not necessarily have to be known. You could model a range of scenarios where the energy lost per collision varies, coming up with a range of time values. The key is relating the time taken to reach rest (call it [itex] t_r [/itex]) with the energy lost per collision (call it [itex] E_c [/itex]) the initial energy (call it [itex] E_0 [/itex]) and the frequency of the pendulum (call it [itex] \omega [/itex]) so you get something like this;

In the such case, if you use at least 5 beads or marbles the amount you pull back should show more time unless you use all but 1. In that case, it will act as you pulled only one bead back due to the loss of kinetic energy. If you use an even amount of beads, try to pull back half. I am doing this precise project for a science fair. I will try to record the information I gather. Hope it helps.